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Algebraic structure
An algebraic structure is one or more sets combined with one or more operations, and optionally with a relation (usually a total order) that satisfies a given set of properties. A magma (or groupoid) is a set with a single closed binary operation (addition). ::a + b = c A semigroup is a magma where the addition is associative. ::a + (b + c) = (a + b) + c A monoid is a semigroup with an additive identity element. ::a + 0 = a A group is a monoid with additive inverse elements. ::a + (-a) = 0 An abelian group is a group where the addition is commutative. ::a + b = b + a A pseudo-ring is an abelian group that has a second closed, associative, binary operation (multiplication) as well. ::a * (b * c) = (a * b) * c :And these two operations satisfy a distribution law. ::a(b + c) = ab + ac A ring is a pseudo-ring that has a multiplicative identity (see below) ::a * 1 = a A commutative ring is a ring where multiplication commutes, ::a * b = b * a A field is a commutative ring where every element has a multiplicative inverse (and thus there is a multiplicative identity), ::a * (1/a) = 1 :The existence of a multiplicative inverse for every nonzero element automatically implies that there are no zero divisors in a field ::if ab=0 for some a≠0, then we must have b=0 (we call this having no zero-divisors). People study these, and maps between them, because it is stunning how often things can be given a group or ring-like structure. So knowing how these things behave carries a lot of information about many things.﻿ multiplicative identity :From Wikipedia:Ring (mathematics) Multiplicative identity: Fraenkel required a ring to have a multiplicative identity 1, whereas Noether did not. Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a 1. Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of ring, especially in advanced books by notable authors such as Artin, Atiyah and MacDonald, Bourbaki, Eisenbud, and Lang. But even today, there remain many books that do not require a 1. Faced with this terminological ambiguity, some authors have tried to impose their views, while others have tried to adopt more precise terms. In the first category, we find for instance Gardner and Wiegandt, who argue that if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable." In the second category, we find authors who use the following terms: Rings with multiplicative identity: unital ring, unitary ring, ring with unity, ring with identity, or ring with 1 rings not requiring multiplicative identity: rng or pseudo-ring.﻿ =Examples= *Magma *Group **Abelian group **Cyclic group *Ring **Commutative ring **Integral domain **Field *Vector space **Algebra (structure) Category:Algebra